Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{a^2 - 36}{a + 6}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = a$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $t = \dfrac{({a} + {6})({a} {-6})} {a + 6} $ We can divide the numerator and denominator by $(a + 6)$ on condition that $a \neq -6$ Therefore $t = a - 6; a \neq -6$